Finite Group Databasebeta

← #37545253

Subgroups — #37545253

Order 23168.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153844C211yesmaximal, characteristic, complemented
2C2×#19153839C211yesmaximal, characteristic, complemented
3#26041931C211yesmaximal, characteristic, complemented
4#26041920C211yesmaximal, characteristic, complemented
5#26041672C211yesmaximal, characteristic, complemented
6#26041509C211yesmaximal, characteristic, complemented
7#26041497C211yesmaximal, characteristic, complemented
8#26041483C211yesmaximal, characteristic, complemented
9#26041444C211yesmaximal, characteristic, complemented
10#26041439C211yesmaximal, characteristic, complemented
11#26041429C211yesmaximal, characteristic, complemented
12#26041427C211yesmaximal, characteristic, complemented
13#26041390C211yesmaximal, characteristic, complemented
14#26041348C211yesmaximal, characteristic, complemented
15#470×C181C211yesmaximal, characteristic, complemented
16C2×#15668450C2211yescharacteristic
17C2×#15668450C2211yescharacteristic, complemented
18C2×#15668449C2211yescharacteristic
19C22×D724C2211yescharacteristic
20#19153844—24no—
21#19153839—24no—
22C2×#15668439C2211yescharacteristic, complemented
23C2×#15668438C2211yescharacteristic
24C2×#15668438C2211yescharacteristic
25C2×#15668436C2211yescharacteristic
26C2×#15668436C2211yescharacteristic
27C2×#15668436C2211yescharacteristic, complemented
28C2×D1448C2211yescharacteristic
29C2×D1448C2211yescharacteristic
30C2×C8×D181C2211yescharacteristic, complemented
31#19153814C2211yescharacteristic, complemented
32#19153805C2211yescharacteristic
33#19153801—21no—
34#19153801—21no—
35#19153798—21no—
36#19153796C2211yescharacteristic, complemented
37#19153791—21no—
38#19153791C2211yescharacteristic
39#19153789—21no—
40#19153786—21no—
41#19153778C2211yescharacteristic
42#19153778C2211yescharacteristic, complemented
43C4×C22⋊D181C2211yescharacteristic
44C4⋊C4×D181—21no—
45C22⋊C4×D181—21no—
46#19153745C2211yescharacteristic, complemented
47#19153739C2211yescharacteristic, complemented
48#19153736C2211yescharacteristic
49#19153736C2211yescharacteristic
50#19153736C2211yescharacteristic, complemented
51C22⋊#13089678C2211yescharacteristic
52#19153728C2211yescharacteristic, complemented
53#19153726C2211yescharacteristic, complemented
54C2×Q16×C181C2211yescharacteristic, complemented
55C2×M16×C181C2211yescharacteristic
56C22⋊Q8×C181C2211yescharacteristic
57C22⋊Q8×C181C2211yescharacteristic
58#106×C181C2211yescharacteristic, complemented
59#103×C181C2211yescharacteristic, complemented
60#103×C181C2211yescharacteristic, complemented
61C23×D181—21no—
62#15668450—22no—
63#15668450—22no—
64#15668450—41no—
65#15668450—41no—
66#15668449—22no—
67#15668449—41no—
68C2×Dic362C2311yescharacteristic
69C2×D724—22no—
70C2×D724—21no—
71C2×D724—21no—
72C2×D724—21no—
73C2×D724—21no—
74C2×D724—41no—
75C2×D724C2311yescharacteristic
76C2×D724C2311yescharacteristic
77C2×D724C2311yescharacteristic
78C2×C22⋊D181—21no—
79C2×C22⋊D181D411yescharacteristic
80C2×C4×D181—21no—
81C2×C4×D181—21no—
82C2×C4×D181—21no—
83C2×C4×D181—21no—
84C2×C4×D181—21no—
85C2×C4×D181C2311yescharacteristic
86#15668439—41no—
87#15668438—22no—
88#15668438—22no—
89#15668436—22no—
90#15668436—22no—
91#15668436—41no—
92D1448—22no—
93D1448D414yescomplemented
94C2×#13089678C2311yescharacteristic
95C2×#13089678C2311yescharacteristic
96C8×D181—22no—
97C4⋊Dic181—21no—
98#15668428—21no—
99#15668428—21no—
100#15668428D411yescharacteristic
101C22⋊Dic181D411yescharacteristic
102#15668426—21no—
103#15668426—21no—
104#15668426—21no—
105#15668426—21no—
106#15668426—21no—
107#15668426—21no—
108#15668426D411yescharacteristic
109C4×Dic181C2311yescharacteristic
110C2×Q8×C181C2311yescharacteristic
111C2×Q8×C181C2311yescharacteristic
112C22×C4×C181C2311yescharacteristic
113Q16×C181—22no—
114M16×C181—22no—
115C2×C8×C181C2311yescharacteristic
116C2×C8×C181C2311yescharacteristic
117C4⋊C4×C181—21no—
118C4⋊C4×C181—21no—
119C4⋊C4×C181C2311yescharacteristic
120C4⋊C4×C181C2311yescharacteristic
121C22⋊C4×C181—21no—
122C22⋊C4×C181—21no—
123C22×D181—21no—
124C22×D181—21no—
125C22×D181—21no—
126C22×D181—42no—
127C22×D181—42no—
128C22×D181—81no—
129C22×D181C2×D411yescharacteristic
130Dic362—21no—
131Dic362—21no—
132D724—21no—
133D724—21no—
134D724—21no—
135D724—21no—
136D724—42no—
137D724—42no—
138D724—42no—
139D724—81no—
140D724C2×D412yes—
141D724C2×D412yes—
142C22⋊D181—41no—
143C22⋊D181—42no—
144C2×Dic181—21no—
145C2×Dic181C2×D411yescharacteristic
146C2×Dic181C2×D411yescharacteristic
147C2×Dic181C2×D411yescharacteristic
148C4×D181—21no—
149C4×D181—21no—
150C4×D181—42no—
151C4×D181—42no—
152C4×D181—42no—
153C4×D181—42no—
154C4×D181—81no—
155#13089678—21no—
156#13089678—21no—
157C23×C181C2×D411yescharacteristic
158Q8×C181—21no—
159Q8×C181—21no—
160Q8×C181—21no—
161Q8×C181—21no—
162C2×C4×C181—21no—
163C2×C4×C181—21no—
164C2×C4×C181—21no—
165C2×C4×C181—21no—
166C2×C4×C181—21no—
167C2×C4×C181—21no—
168C2×C4×C181C2×D411yescharacteristic
169C2×C4×C181C2411yescharacteristic
170C8×C181—21no—
171C8×C181C2×D412yes—
172C22×C181—21no—
173C22×C181—21no—
174C22×C181—21no—
175C22×C181C22×D411yescharacteristic
176C2×D181—21no—
177C2×D181—22no—
178C2×D181—22no—
179C2×D181—21no—
180C2×D181—21no—
181C2×D181—41no—
182C2×D181—41no—
183C2×D181—41no—
184C2×D181—41no—
185C2×D181—41no—
186C2×D181—41no—
187C2×D181—41no—
188C2×D181—84no—
189C4×C181—21no—
190C4×C181—41no—
191C4×C181—41no—
192C4×C181—41no—
193C4×C181—41no—
194C4×C181C22×D411yescharacteristic
195C4×C181H(5,𝔽2)11yescharacteristic
196Dic181—21no—
197Dic181—21no—
198Dic181—21no—
199Dic181—41no—
200C2×C181—41no—
201C2×C181D4211yescharacteristic
202C2×C181#57511yescharacteristic
203C2×C181#57711yescharacteristic
204D181—41no—
205D181—42no—
206D181—42no—
207D181—81no—
208C181#326711yescharacteristic, complemented, Sylow
209#3267—1811nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1811no—
211C2×#135—1811no—
212#547—1811no—
213#546—1811no—
214#497—1811no—
215#470D18111yescharacteristic, complemented
216#465—1811no—
217#463—1811no—
218#461—1811no—
219#452—1811no—
220#452—1811no—
221#450—1811no—
222#449—1811no—
223#442—1811no—
224#435—1811no—
225C2×#41—1811no—
226C2×#41—1811no—
227C2×#41—1811no—
228C22×D4—1811no—
229AGL(1,ℤ/8)—3624no—
230#135—3624no—
231C2×Q16—1811no—
232C2×Q16C2×D18111yescharacteristic, complemented
233C2×SD16—1811no—
234C2×SD16—1811no—
235C2×SD16—1811no—
236C2×SD16—1811no—
237C2×SD16—1811no—
238C2×D8—1811no—
239C2×D8—1811no—
240C2×M16C2×D18111yescharacteristic
241C22×C8—1811no—
242C4⋊Q8—1811no—
243#124—1811no—
244#123—3621no—
245#123—3621no—
246#123—3621no—
247C22⋊Q8—1811no—
248C22⋊Q8C2×D18111yescharacteristic
249C22⋊Q8C2×D18111yescharacteristic
250#121—3621no—
251#121—3621no—
252#121—1811no—
253#120—3621no—
254C4×D4—1811no—
255C4×D4—1811no—
256C4×D4—1811no—
257C2×C4⋊C4—3621no—
258C2×C22⋊C4—3621no—
259#106C2×D18111yescharacteristic, complemented
260C4⋊C8—1811no—
261#103—1811no—
262#103C2×D18111yescharacteristic, complemented
263#103C2×D18111yescharacteristic, complemented
264#102—1811no—
265#102—1811no—
266#102—1811no—
267C22⋊C8—1811no—
268C22⋊C8—1811no—
269#97—1811no—
270C24—3621no—
271#41—3622no—
272#41—3622no—
273#41—3622no—
274#41—7241no—
275#41—7241no—
276#41—7241no—
277C2×Q8—1811no—
278C2×Q8C22×D18111yescharacteristic
279C2×Q8C22×D18111yescharacteristic
280C2×D4—3622no—
281C2×D4—3621no—
282C2×D4—3621no—
283C2×D4—3621no—
284C2×D4—3621no—
285C2×D4—3621no—
286C2×D4—1811no—
287C2×D4—1811no—
288C2×D4—1811no—
289C2×D4—1811no—
290C2×D4—7241no—
291C22×C4—3621no—
292C22×C4—3621no—
293C22×C4—3621no—
294C22×C4—3621no—
295C22×C4—3621no—
296C22×C4—1811no—
297C22×C4C22×D18111yescharacteristic
298Q16—22no—
299Q16—7241no—
300SD16—3622no—
301SD16—3622no—
302SD16—3622no—
303SD16—3622no—
304SD16—7241no—
305D8—3622no—
306D8—1814no—
307M16—22no—
308C2×C8—3622no—
309C2×C8—1811no—
310C2×C8—1811no—
311C2×C8C22×D18111yescharacteristic
312C2×C8C22×D18111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—3621no—
316C4⋊C4—3621no—
317C4⋊C4—3621no—
318C4⋊C4—1811no—
319C4⋊C4C22×D18111yescharacteristic
320C4⋊C4C22×D18111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—3621no—
324C22⋊C4—3621no—
325C22⋊C4—3621no—
326C22⋊C4—3621no—
327C22⋊C4—3621no—
328C22⋊C4—3621no—
329C22⋊C4—1811no—
330C22⋊C4—1811no—
331C42—1811no—
332C23—3621no—
333C23—3621no—
334C23—3621no—
335C23—14481no—
336C23—1811no—
337C23—7242no—
338C23—7242no—
339C23D4×D18111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—3621no—
345Q8—3621no—
346D4—3621no—
347D4—3621no—
348D4—3621no—
349D4—3621no—
350D4—14481no—
351D4—1812no—
352D4—1812no—
353D4—7242no—
354D4—7242no—
355D4—7242no—
356D4—7241no—
357D4—7242no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3621no—
365C2×C4—3621no—
366C2×C4—3621no—
367C2×C4—14481no—
368C2×C4—1811no—
369C2×C4—1811no—
370C2×C4—1811no—
371C2×C4—7242no—
372C2×C4—7242no—
373C2×C4—7242no—
374C2×C4—7242no—
375C2×C4D4×D18111yescharacteristic
376C2×C4C23×D18111yescharacteristic
377C8—21no—
378C8—3621no—
379C8—3621no—
380C8D4×D18112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—3621no—
385C22—3622no—
386C22—3622no—
387C22—3621no—
388C22—3621no—
389C22—14484no—
390C22—7241no—
391C22—7241no—
392C22—7241no—
393C22—7241no—
394C22—7241no—
395C22—7241no—
396C22—7241no—
397C22C2×D4×D18111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3621no—
404C4—3621no—
405C4—3621no—
406C4—7241no—
407C4C2×D4×D18111yescharacteristic
408C4#1915386211yescharacteristic
409C2—41no—
410C2—14481no—
411C2—7241no—
412C2—7242no—
413C2—7242no—
414C2#2604191311yescharacteristic
415C2#2604207811yescharacteristic
416C2#2604208811yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#470×C181C2
socleC22×C181C22×D4
hypercenter#470D181
hypocenterC181#3267
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153844C2semidirect
2. C2×#19153839C2semidirect
3. #26041931C2semidirect
4. #26041920C2semidirect
5. #26041672C2semidirect
6. #26041509C2semidirect
7. #26041497C2semidirect
8. #26041483C2semidirect
9. #26041444C2semidirect
10. #26041439C2semidirect
11. #26041429C2semidirect
12. #26041427C2semidirect
13. #26041390C2semidirect
14. #26041348C2semidirect
15. #470×C181C2semidirect
17. C2×#15668450C22semidirect
22. C2×#15668439C22semidirect
27. C2×#15668436C22semidirect
30. C2×C8×D181C22semidirect
31. #19153814C22semidirect
36. #19153796C22semidirect
42. #19153778C22semidirect
46. #19153745C22semidirect
47. #19153739C22semidirect
50. #19153736C22semidirect
52. #19153728C22semidirect
53. #19153726C22semidirect
54. C2×Q16×C181C22semidirect
58. #106×C181C22semidirect
59. #103×C181C22semidirect
60. #103×C181C22semidirect
93. D1448D4semidirect
208. C181#3267semidirect
215. #470D181semidirect
232. C2×Q16C2×D181semidirect
259. #106C2×D181semidirect
262. #103C2×D181semidirect
263. #103C2×D181semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191538442
C2×#191538392
#260419312
#260419202
#260416722
#260415092
#260414972
#260414832
#260414442
#260414392
#260414292
#260414272
#260413902
#260413482
#470×C1812
#3267181